Increasing and decreasing piecewise function on an interval

I'm working on a problem that involves finding the intervals where a function f is increasing and decreasing. Given the function $f(x)=\{\begin{array}{ll}x+7& {\textstyle \text{text{if} xlt -3}}\\ |x+1|& {\textstyle \text{text{if} -3le x 1}}\\ 5-2x& {\textstyle \text{text{if} xge 1}}\end{array}$

I worked out that f is increasing on $(-\mathrm{\infty},-3)$ and $[-1,1)$, and f is decreasing on $[-3,-1],[1,\mathrm{\infty})$.

However, the solution in the book claims that f is increasing on [-1,1], rather than [-1,1) like I worked out. I am having trouble understanding why the book claims that this is the case.

I was under the impression that f must be differentiable on the interior of an interval I and continuous on all of I in order to make any statements about increasing/decreasing behavior on the closed interval I. I'm not sure if I am overlooking something, but it seems that f is not continuous on [-1,1]. I would appreciate any clarification.

I'm working on a problem that involves finding the intervals where a function f is increasing and decreasing. Given the function $f(x)=\{\begin{array}{ll}x+7& {\textstyle \text{text{if} xlt -3}}\\ |x+1|& {\textstyle \text{text{if} -3le x 1}}\\ 5-2x& {\textstyle \text{text{if} xge 1}}\end{array}$

I worked out that f is increasing on $(-\mathrm{\infty},-3)$ and $[-1,1)$, and f is decreasing on $[-3,-1],[1,\mathrm{\infty})$.

However, the solution in the book claims that f is increasing on [-1,1], rather than [-1,1) like I worked out. I am having trouble understanding why the book claims that this is the case.

I was under the impression that f must be differentiable on the interior of an interval I and continuous on all of I in order to make any statements about increasing/decreasing behavior on the closed interval I. I'm not sure if I am overlooking something, but it seems that f is not continuous on [-1,1]. I would appreciate any clarification.